Abstract.
The notion of a graph type \(\mathcal{T}\) is introduced by a collection of axioms. A graph of type \(\mathcal{T}\) (or \(\mathcal{T}\)-graph) is defined as a set of edges, of which the structure is specified by \(\mathcal{T}\). From this, general notions of subgraph and isomorphism of \(\mathcal{T}\)-graphs are derived. A Cantor-Bernstein (CB) result for \(\mathcal{T}\)-graphs is presented as an illustration of a general proof for different types of graphs. By definition, a relation \(\mathcal{R}\) on \(\mathcal{T}\)-graphs satisfies the CB property if \(A \mathcal{R} B\) and \(B \mathcal{R} A\) imply that A and B are isomorphic. In general, the relation ‘isomorphic to a subgraph’ does not satisfy the CB property. However, requiring the subgraph to be disconnected from the remainder of the graph, a relation that satisfies the CB property is obtained. A similar result is shown for \(\mathcal{T}\)-graphs with multiple edges.
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Received: 25 October 1996 / 5 February 1998
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Engelfriet, J., Gelsema, T. Axioms for generalized graphs, illustrated by a Cantor–Bernstein proposition. Acta Informatica 35, 1075–1096 (1998). https://doi.org/10.1007/s002360050152
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DOI: https://doi.org/10.1007/s002360050152